walters 3 years ago Question based on (Groups)

1. walters

jjj let the mapping $\tau _{ab}$ for a,b element R ,maps the reals by the ruel $\tau _{ab}:x \rightarrow ax+b.Let G={\tau _{ab}:a \neq 0}$ Determine whether or not G is an abelian group under the composition of mappings

2. walters

oops ignore jjj

3. KingGeorge

Well, let $$\tau_{ab},\tau_{cd}\in G$$. Then $\tau_{ab}\circ\tau_{cd}(x)=\tau_{ab}(cx+d)=acx+ad+b.$Also, $\tau_{cd}\circ\tau_{ab}(x)=\tau_{cd}(ax+b)=cax+bc+d.$Since $$acx+ad+b\neq acx+bc+d$$, it must be that $$G$$ is not abelian.

4. walters

am i not suppose to check whether is a group using the condition before i can conclude that is an abelian group

5. KingGeorge

It just asked if it was an abelian group. Since the elements don't commute, it can't be an abelian group even if it is a group. However, if we want to check it's a group, we can do that as well. Have you shown any of the axioms for proving it's a group yet?

6. walters

i am failing to show the axioms

7. KingGeorge

First, we can see that $$G$$ is closed under composition. $\tau_{ab}\circ\tau_{cd}(x)=\tau_{ab}(cx+d)=acx+ad+b=\tau_{(ac)(ad+b)}.$We also have the identity $$\tau_{10}$$. Now we just have to check associativity and inverses.

8. walters

why do u chose to use |dw:1360875075213:dw|

9. KingGeorge

"$$\circ$$" is a relatively common notation to denote the composition of two functions. You can also use $$\tau_{ab}(\tau_{cd}(x))$$ is you prefer.

10. walters

can u also use * if u wnt

11. KingGeorge

For inverses, we want $\tau_{ab}\circ\tau_{cd}(x)=acx+ad+b=x$for some $$c,d\in\mathbb{R}$$. We can see that $$c=a^{-1}$$ and $$d=a^{-1}(-b)$$ work for this direction. For the other, $\tau_{cd}\circ\tau_{ab}(x)=cax+bc+d=a^{-1}ax+ba^{-1}-a^{-1}b=x,$so we do indeed have inverses.

12. KingGeorge

If you make it clear your group function is composition of functions, you can use * as well.

13. KingGeorge

As for associativity, I'm just going to use the fact that composition of functions is always associative. If you have not heard of that yet, we can still go over how to prove this particular function is associative.

14. walters

15. KingGeorge

let $$\tau_{ab},\tau_{cd},\tau_{ef}\in G$$. Then$\tau_{ab}\circ(\tau_{cd}\circ\tau_{ef})(x)=\tau_{ab}=\tau_{ab}(cex+cf+d)=acex+acf+ad+b$and$(\tau_{ab}\circ\tau_{cd})\circ\tau_{ef}(x)=(\tau_{(ac)(ad+b)})(ex+f)=acex+acf+ad+b$Since these are equal, it's associative, and we have a group.

16. walters

k it is a group but not an abelian group

17. KingGeorge

Yup.

18. walters

thnx

19. KingGeorge

You're welcome.

20. walters

can we verify identity

21. KingGeorge

$\tau_{10}(x)=1x+0=x$